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Large Time existence For 1D Green-Naghdi equations
Samer Israwi
To cite this version:
EQUATIONS
SAMER ISRAWI
Abstract. We consider here the 1D Green-Naghdi equations that are com-monly used in coastal oceanography to describe the propagation of large ampli-tude surface waves. We show that the solution of the Green-Naghdi equations can be constructed by a standard Picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition.
1. Introduction
1.1. Presentation of the problem. The water-waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect, incompressible, irrotational fluid under the influence of gravity. This motion is described by the free surface Euler equations that are known to be well-posed after the works of Nalimov [16], Yasihara [21], Craig [6], Wu [19, 20] and Lannes [11]. But, because of the complexity of these equations, they are often replaced for pratical purposes by approximate asymptotic systems. The most prominent examples are the Green-Naghdi equations (GN) – which is a widely used model in coastal oceanography ([8, 4, 7] and, for instance, [18, 10])–, the Shallow-Water equations, and the Boussinesq systems; their range of validity depends on the physical characteristics of the flow under consideration. In other words, they depend on certain assumptions made on the dimensionless parameters ε, µ defined as: ε = a h0 , µ =h 2 0 λ2;
where a is the order of amplitude of the waves and the bottom variations; λ is the wave-length of the waves and the bottom variations; h0is the reference depth.
The parameter ε is often called nonlinearity parameter; while µ is the shallowness parameter. In the shallow-water scaling (µ ≪ 1), and without smallness assumption on ε one can derive the so-called Green-Naghdi equations (see [8, 13] for a derivation and [2] for a rigorous justification) also called Serre or fully nonlinear Boussinesq equations [15].
In nondimensionalized variables, denoting by ζ(t, x) and u(t, x) the parameteriza-tion of the surface and the vertically averaged horizontal component of the velocity
Ce travail a b´en´efici´e d’une aide de l’Agence Nationale de la Recherche portant la r´ef´erence ANR-08-BLAN-0301-01.
at time t, and by b(x) the parameterization of the bottom, the equations read (1) ∂tζ + ∇ · (hu) = 0, (h + µhT [h, εb])∂tu + h∇ζ + εh(u · ∇)u + µε −1 3∇[(h 3 ((u · ∇)(∇ · u) − (∇ · u)2)] + hℜ[h, εb]u = 0, where h = 1 + ε(ζ − b) and T [h, εb]W = − 1 3h∇(h 3∇ · W ) + ε 2h[∇(h 2∇b · W ) − h2∇b∇ · W ] + ε2∇b∇b · W,
while the purely topographical term ℜ[h, εb]u is defined as: ℜ[h, εb]u = ε
2h[∇(h
2(u · ∇)2b) − h2((u · ∇)(∇ · u) − (∇ · u)2)∇b]
+ε2((u · ∇)2b)∇b.
This model is often used in coastal oceanography because it takes into account the dispersive effects neglected by the shallow-water and it is more nonlinear than the Boussinesq equations. A recent rigorous justification of the GN model was given by Li [14] in 1D and for flat bottoms, and by B. Alvarez-Samaniego and D. Lannes [2] in 2008 in the general case. This latter reference relies on well-posedness results for these equations given in [3] and based on general well-posedness results for evolution equations using a Nash-Moser scheme. The result of [3] covers both the case of 1D and 2D surfaces, and allows for non flat bottoms. The reason why a Nash-Moser scheme is used there is because the estimates on the linearized equations exhibit losses of derivatives. However, in the 1D case with flat bottoms, such losses do not occur and it is possible to construct a solution with a standard Picard iterative scheme as in [14]. Our goal here is to show that it is also possible to use such a simple scheme in the 1D case with non flat bottoms, thanks to a careful analysis of the linearized equations.
1.2. Organization of the paper. We start by giving some preliminary results in Section 2.1; the main theorem is then stated in Section 2.2 and proved in Section 2.3. Finally, in Appendix A, we give the existence and uniqueness of a solution to the linear Cauchy problem associated to the Green-Naghdi equations. The proof of the energy conservation, stated in the main theorem, is given in Appendix B. 1.3. Notation. We denote by C(λ1, λ2, ...) a constant depending on the parameters
λ1, λ2, ... and whose dependence on the λj is always assumed to be nondecreasing.
The notation a . b means that a ≤ Cb, for some nonegative constant C whose exact expression is of no importance (in particular, it is independent of the small
parameters involved ).
Let p be any constant with 1 ≤ p < ∞ and denote Lp = Lp(R) the space of all
Lebesgue-measurable functions f with the standard norm |f|Lp=
Z
R
|f(x)|dx1/p < ∞.
When p = 2, we denote the norm | · |L2 simply by | · |2. The inner product of any
functions f1and f2in the Hilbert space L2(R) is denoted by
(f1, f2) =
Z
R
The space L∞ = L∞(R) consists of all essentially bounded, Lebesgue-measurable
functions f with the norm
|f|L∞ = ess sup |f(x)| < ∞.
We denote by W1,∞= W1,∞(R) = {f ∈ L∞, ∂
xf ∈ L∞} endowed with its
canoni-cal norm.
For any real constant s, Hs= Hs(R) denotes the Sobolev space of all tempered
dis-tributions f with the norm |f|Hs= |Λsf |2< ∞, where Λ is the pseudo-differential
operator Λ = (1 − ∂x2)1/2.
For any functions u = u(x, t) and v(x, t) defined on R × [0, T ) with T > 0, we denote the inner product, the Lp-norm and especially the L2-norm, as well as the
Sobolev norm, with respect to the spatial variable x, by (u, v) = (u(·, t), v(·, t)), |u|Lp= |u(·, t)|Lp, |u|L2 = |u(·, t)|L2 , and |u|Hs= |u(·, t)|Hs, respectively.
Let Ck(R) denote the space of k-times continuously differentiable functions and
C∞
0 (R) denote the space of infinitely differentiable functions, with compact
sup-port in R; we also denote by C∞
b (R) the space of infinitely differentiable functions
that are bounded together with all their derivatives.
Let f be a function of the independent variables x1, x2,...,xm; its partial derivative
with respect to xk is denoted by ∂xkf = fxk for 1 ≤ k ≤ m.
For any closed operator T defined on a Banach space X of functions, the commu-tator [T, f ] is defined by [T, f ]g = T (f g) − fT (g) with f, g and fg belonging to the domain of T .
2. Well-posedness of the Green-Naghdi equations in 1D
For one dimensional surfaces, the Green-Naghdi equations (1) can be simplified, after some computations, into
(2) ( ∂tζ + ∂x(hu) = 0, (h + µhT [h, εb])[∂tu + εu∂xu] + h∂xζ + εµhQ[h, εb](u) = 0 where h = 1 + ε(ζ − b) and T [h, εb]w = − 1 3h∂x(h 3w x) + ε 2h[∂x(h 2b xw) − h2bxwx] + ε2b2xw, Q[h, εb](w) = 2 3h∂x(h 3w2 x) + εhwx2bx+ ε 1 2h∂x(h 2w2b xx) + ε2w2bxxbx.
Remark 1. The interest of the formulation (2) of the Green-Naghdi equation is that all the third order derivatives of u have been factorized by (h + µhT [h, εb]). Indeed, Q[h, εb] is a second order differential operator. This was used in [14] in the case of flat bottoms (b = 0).
2.1. Preliminary results. For the sake of simplicity, we write T= h + µhT [h, εb].
We always assume that the nonzero depth condition
(3) ∃ h0> 0, inf
x∈Rh ≥ h0, h = 1 + ε(ζ − b)
is valid initially, which is a necessary condition for the GN system (2) to be phys-ically valid. We shall demonstrate that the operator T plays an important role in
the energy estimate and the local well-posedness of the GN system (2). Therefore, we give here some of its properties.
The following lemma gives an important invertibility result on T.
Lemma 1. Let b ∈ Cb∞(R) and ζ ∈ W1,∞(R) be such that (3) is satisfied. Then
the operator
T: H2(R) −→ L2(R)
is well defined, one-to-one and onto.
Remark 2. Here and throughout the rest of this paper, and for the sake of sim-plicity, we do not try to give some optimal regularity assumption on the bottom parameterization b. This could easily be done, but is of no interest for our present purpose. Consequently, we ommit to write the dependance on b of the different quantities that appear in the proof.
Proof. In order to prove the invertibility of T, let us first remark that the quantity
|v|2
∗ defined as
|v|2∗= |v|22+ µ|∂xv|22,
is equivalent to the H1(R)-norm but not uniformly with respect to µ ∈ (0, 1). We
define by H1
∗(R) the space H1(R) endowed with this norm. The bilinear form:
a(u, v) = (hu, v) + µ h √h 3ux− √ 3 2 εbxu, h √ 3vx− √ 3 2 εbxv + µε2 4 (hbxu, bxv). is obviously continous on H1 ∗(R) × H∗1(R). Remarking that a(v, v) = (Tv, v) = (hv, v) +µ h √h 3vx− √ 3 2 εbxv, h √ 3vx− √ 3 2 εbxv + µε2 4 (hbxv, bxv), we have |v|2∗ ≤ |v|22+ 3µ h2 0| h √ 3vx| 2 2 ≤ |v|22+ 6µ h2 0 |√h 3vx− √ 3 2 εbxv| 2 2+ 3ε2 4 |bxv| 2 2 . One deduces that
maxn1,18 h2 0 o |v|22+ µ| h √ 3vx− √ 3 2 εbxv| 2 2+ µε2 4 |bxv| 2 2 ≥ |v|2∗.
Since from (3) we also get
a(v, v) ≥ h0|v|22+ µh0 |√h 3vx− √ 3 2 εbxv| 2 2+ µε2 4 |bxv| 2 2 , it is easy to deduce that
a(v, v) ≥ h0 max1,18 h2 0 |v| 2 ∗. (4) In particular, a is coercive on H1
∗. Using Lax-Milgram lemma, for all f ∈ L2(R),
there exists unique u ∈ H1
∗(R) such that, for all v ∈ H∗1(R)
equivalently, there is a unique variational solution to the equation
(5) Tu = f.
We then get from the definition of T that ∂x2u =
hu +εµ2∂x(h2bx)u + ε2µhb2xu −µ3∂xh3∂xu − f µh3
3
; since u ∈ H1(R) and f ∈ L2(R), we get ∂2
xu ∈ L2(R) and thus u ∈ H2(R).
The following lemma then gives some properties of the inverse operator T−1.
Lemma 2. Let b ∈ Cb∞(R), t0 > 1/2 and ζ ∈ Ht0+1(R) be such that (3) is
satis-fied. Then: (i) ∀0 ≤ s ≤ t0+ 1, |T−1f |Hs+ √µ|∂xT−1f |Hs ≤ C(1 h0, |h − 1|Ht0+1)|f|H s; (ii) ∀0 ≤ s ≤ t0+ 1, √µ|T−1∂xg|Hs ≤ C(1 h0, |h − 1|Ht0 +1)|g|H s;
(iii) If s ≥ t0+ 1 and ζ ∈ Hs(R) then:
k T−1 k
Hs(R)→Hs(R)+√µ k T−1∂xkHs(R)→Hs(R)≤ cs,
where csis a constant depending on h1
0, |h−1|H
sand independent of (µ,ε) ∈ (0, 1)2.
Proof. Step 1. We prove that if u ∈ H1
∗(R) solves
Tu = f +√µ∂xg for f, g ∈ L2(R), then one has
|u|H1 ∗ ≤ C 1 h0 |f|2+ |g|2.
Indeed, multiplying the equation by u and integrating by parts, one gets, with the notations used in the proof of lemma 1
a(u, u) ≤ (f, u) − (g,√µ∂xu).
We thus get from the proof of Lemma 1 and Cauchy-Schwarz inequality that h0 max{1,h182 0} |u|2H1 ∗ ≤ |f|2|u|2+ |g|2|u|H 1 ∗,
and the result follows easily.
Step 2. We prove here that |T−1f |Hs+ √µ|∂xT−1f |Hs ≤ C(1
h0, |h − 1|Ht0+1)|f|Hs.
Indeed, if f ∈ Hsand u = T−1f then Tu = f . Applying Λsto this identity, we get
T(Λsu) = Λsf + [T, Λs]u = f +˜ √µ∂xg,˜ with, ˜ f = Λsf − [Λs, h]u +εµ 2 [Λ s, h2b x]ux− ε2µ[Λs, h2bx]u, and ˜ g = √µ 3 [Λ s, h3]u x− ε√µ 2 [Λ s, h2b x]u.
Now, one can deduce from the commutator estimate (see e.g Lemma 4.6 of [3]) (6) |[Λs, F ]G|2.|∇F |Ht0|G|Hs−1 that | ˜f |2+ |˜g|2 ≤ |f|Hs+ C 1 h0, |h − 1|H t0 +1 |u|Hs−1+√µ|∂xu|Hs−1.
One can use Step 1 and a continuous induction on s to show that the inequality (i) holds for 0 ≤ s ≤ t0+ 1.
Step 3. We prove here that √µ|T−1∂
xg|Hs ≤ C(1
h0, |h − 1|Ht0 +1)|g|Hs. Indeed, if
g ∈ Hsand u = √µT−1∂
xg then Tu = √µ∂xg and thus
T(Λsu) = ˜f +√µ∂x˜g, with, ˜ f = −[Λs, h]u +εµ 2 [Λ s, h2b x]ux− ε2µ[Λs, h2bx]u, and ˜ g = Λsg + √µ 3 [Λ s, h3]u x− ε√µ 2 [Λ s, h2b x]u.
Proceeding now as for the Step 2, one can deduce (ii).
Step 4. If s ≥ t0+ 1 then one can prove (iii) proceeding as in Step 2 and 3 above,
but replacing the commutator estimate (6) by the following one
(7) |[Λs, F ]G|
2.|∇F |Hs−1|G|Hs−1.
2.2. Linear analysis. In order to rewrite the GN equations (2) in a condensed form, let us decompose Q[h, εb](u) as
εµhQ[h, εb](u) = Q1[U ]ux+ q2(U ) where U = (ζ, u)T and Q1[U ]f = 2 3εµ∂x(h 3u xf ) + ε2µh2bxuxf + ε2µh2bxxuf (8) q2(U ) = ε3µhbxxbxu2+ 1 2ε 2µ∂ x(h2bxx)u2.
The Green-Naghdi equations (2) can be written after applying T−1 to both sides
of the second equation in (2) as
(9) ∂tU + A[U ]∂xU + B(U ) = 0,
with U = (ζ, u)T and where
(10) A[U ] = εu h T−1(h·) εu + T−1Q1[U ]
and (11) B(U ) = εbxu T−1q 2(U ) .
This subsection is devoted to the proof of energy estimates for the following initial value problem around some reference state U = (ζ, u)T:
(12)
∂tU + A[U ]∂xU + B(U ) = 0;
U|t=0 = U0.
We define now the Xs spaces, which are the energy spaces for this problem.
Definition 1. For all s ≥ 0 and T > 0, we denote by Xsthe vector space Hs(R) ×
Hs+1(R) endowed with the norm
∀ U = (ζ, u) ∈ Xs, |U|2Xs := |ζ|2Hs+ |u|2Hs+ µ|∂xu|2Hs,
while Xs
T stands for C([0,Tε]; Xs) endowed with its canonical norm.
First remark that a symmetrizer for A[U ] is given by
(13) S = 1 0 0 T ,
with h = 1 + ε(ζ− b) and T = h + µhT [h, εb]. A natural energy for the IVP (12) is given by
(14) Es(U )2= (ΛsU, SΛsU ).
The link between Es(U ) and the Xs-norm is investigated in the following Lemma.
Lemma 3. Let b ∈ Cb∞(R), s ≥ 0 and ζ ∈ W1,∞(R). Under the condition (3), Es(U ) is uniformly equivalent to the | · |
Xs-norm with respect to (µ, ε) ∈ (0, 1)2:
Es(U ) ≤ C |h|L∞, |hx|L∞|U|Xs,
and
|U|Xs ≤ C 1
h0
Es(U ).
Proof. Notice first that
Es(U )2= |Λsζ|22+ (Λsu, TΛsu),
one gets the first estimate using the explicit expression of T, integration by parts and Cauchy-Schwarz inequality.
The other inequality can be proved by using that infx∈Rh ≥ h0> 0 and proceeding
as in the proof of Lemma 1.
We prove now the energy estimates in the following proposition:
Proposition 1. Let b ∈ Cb∞(R), t0 > 1/2, s ≥ t0+ 1. Let also U = (ζ, u)T
all U0 ∈ Xs there exists a unique solution U = (ζ, u)T ∈ XTs to (12) and for all 0 ≤ t ≤Tε Es(U (t)) ≤ eελTtEs(U 0) + ε Z t 0 eελT(t−t′)C(Es(U )(t′))dt′.
For some λT = λT(sup0≤t≤T /εEs(U (t)), sup0≤t≤T /ε|∂th(t)|L∞) .
Proof. Existence and uniqueness of a solution to the IVP (12) is achieved in
ap-pendix A and we thus focus our attention on the proof of the energy estimate. For any λ ∈ R, we compute eελt∂ t(e−ελtEs(U )2) = −ελEs(U )2+ ∂t(Es(U )2). Since Es(U )2= (ΛsU, SΛsU ), we have (15) ∂t(Es(U )2) = 2(Λsζ, Λsζt) + 2(Λsu, TΛsut) + (Λsu, [∂t, T]Λsu).
One gets using the equations (12) and integrating by parts, 1 2e ελt∂ t(e−ελtEs(U )2) = −ελ 2 E s(U )2 − (SA[U]Λs∂xU, ΛsU ) − Λs, A[U ]∂ xU, SΛsU − (ΛsB(U ), SΛsU ) +1 2(Λ su, [∂ t, T]Λsu). (16)
We now turn to bound from above the different components of the r.h.s of (16). • Estimate of (SA[U ]Λs∂ xU, ΛsU ). Remarking that SA[U] = εu h h T(εu·) + Q1[U ] , we get (SA[U ]Λs∂xU, ΛsU ) = (εuΛsζx, Λsζ) + (hΛsux, Λsζ) +(hΛsζx, Λsu) + (T(εu·) + Q1[U ])Λsux, Λsu =: A1+ A2+ A3+ A4.
We now focus to control (Aj)1≤j≤4.
− Control of A1. Integrating by parts, one obtains
A1= (εuΛsζ, Λsζx) = −
1 2(εuxΛ
sζ, Λsζ)
one can conclude by Cauchy-Schwarz inequality that |A1| ≤ εC(|ux|L∞)Es(U )2.
− Control of A2+ A3. First remark that
we get,
|A2+ A3| ≤ εC(|hx|L∞)Es(U )2.
− Control of A4. One computes,
A4 = ε T(uΛsux), Λsu + (Q1[U]Λsux, Λsu) =: A41+ A42. Note that A41 = ε(h uΛsux, Λsu) + εµ 3 (h 3(uΛsu x)x, Λsux) −ε 2µ 2 (h 2b x(uΛsux)x, Λsu) −ε 2µ 2 (h 2b xuΛsux, Λsux) +ε3µ(h ub2xΛsux, Λsu); since (h3(uΛsux)x, Λsux) = 1 2 − (h 3 xuΛsux, Λsux) + (h3uxΛsux, Λsux),
by using successively integration by parts and the Cauchy-Schwarz inequality, one obtains directly:
|A41| ≤ εC(|u|W1,∞, |ζ|W1,∞)Es(U )2.
For A42, remark that
|A42| = |(Q1[U ]Λsux, Λsu)| = − 2 3εµ(h 3 uxΛsux, Λsux) + ε2µ(h2uxbxΛsux, Λsu) +ε2µ(h2ub xxΛsux, Λsu) therefore |A42| ≤ εC(|u|W1,∞, |ζ|W1,∞)Es(U )2.
This shows that
|A4| ≤ εC(|u|W1,∞, |ζ|W1,∞)Es(U )2.
• Estimate of Λs, A[U ]∂
xU, SΛsU. Remark first that
Λs, A[U ]∂ xU, SΛsU = ([Λs, εu]ζx, Λsζ) + ([Λs, h]ux, Λsζ) +([Λs, T−1 h]ζx, TΛsu) + ([Λs, εu]ux, TΛsu) + Λs, T−1 Q1[U ]ux, TΛsu =: B1+ B2+ B3+ B4+ B5.
− Control of B1+ B2 = ([Λs, εu]ζx, Λsζ) + ([Λs, h]ux, Λsζ). Since s ≥ t0+ 1, we
can use the commutator estimate (7) to get
− Control of B4= ([Λs, εu]ux, TΛsu). By using the explicit expression of T we get B4 = ([Λs, εu]ux, hΛsu) +µ 3(∂x[Λ s, εu]u x, h3Λsux) −εµ2 ([Λs, εu]ux, h2bxΛsux) +εµ 2 ([Λ s, εu]u x, ∂x(h2bxΛsu)) +ε2µ([Λs, εu]ux, h b2xΛsu),
using the Cauchy-Schwarz inequality and the fact that ∂x[Λs, f ]g = [Λs, fx]g + [Λs, f ]gx
one obtains directly:
|B4| ≤ εC(Es(U ))Es(U )2.
− Control of B3= ([Λs, T−1 h]ζx, TΛsu). Remark first that
T[Λs, T−1]hζx= T[Λs, T−1h]ζx− [Λs, h]ζx; morever, since [Λs, T−1
] = −T−1[Λs, T]T−1
, one gets
T[Λs, T−1h]ζx= −[Λs, T]T−1hζx+ [Λs, h]ζx, and one can check by using the explicit expression of T that
T[Λs, T−1h]ζx = −[Λs, h]T−1hζx+µ 3∂x{[Λ s, h3 ]∂x(T−1hζx)} −εµ2 ∂x[Λs, h2bx]T−1hζx+εµ 2 [Λ s, h2 bx]∂xT−1hζx −ε2µ[Λs, hb2 x]T −1hζ x+ [Λs, h]ζx.
One deduces directly from Lemma 2, an integration by parts, and Cauchy-Schwarz inequality that |B3| ≤ C 1 h0, |h − 1| Hs n |hx|Hs−1 + εµ 2 |h 2b x|Hs+ ε2µ|hb2x|Hs |hζx|Hs−1 + √µ 3 |h 3 x|Hs−1+ ε√µ 2 |h 2 bx|Hs |hζx|Hs−1 + |hx|Hs−1|ζx|Hs−1 o |Λsu|H1 ∗. Finally, since
|hζx|Hs−1 ≤ C(Es(U ))Es(U ) and |hb2x|Hs+ |h2bx|Hs ≤ C(Es(U )),
we deduce
|B3| ≤ εC(Es(U ))Es(U )2.
− Control of B5= Λs, T−1Q1[U ]ux, TΛsu. Let us first write
so, that TΛs, T−1Q1[U ]ux = −[Λs, h]T−1Q1[U ]ux+µ 3∂x{[Λ s, h3 ]∂x(T−1Q1[U ]ux)} −εµ2 ∂x{[Λs, h2bx]T−1Q1[U]ux} + εµ 2 [Λ s, h2 bx]∂x(T−1Q1[U ]ux) −ε2µ[Λs, hb2 x]T−1Q1[U ]ux+ [Λs, Q1[U ]]ux.
To control the term Λs, Q
1[U ]ux, Λsu we use the explicit expression of Q1[U ]:
Q1[U]f =
2 3εµ∂(h
3u
xf ) + ε2µh2bxuxf + ε2µh2bxxuf,
and the fact that
∂x[Λs, f ]g = [Λs, ∂x(f ·)]g.
Similarly to control the term ∂x{[Λs, h3]∂x(T−1Q1[U ]ux)}, Λsu we use the explicit
expression of Q1[U ], the commutator estimate (7) and Lemma 2. Indeed,
∂x{[Λs, h3]∂x(T−1Q1[U ]ux)}, Λsu = −23εµ [Λs, h3]∂x(T−1∂x(h3uxux)), Λsux −ε2µ [Λs, h3]∂x(T−1(h2bxuxux)), Λsux −ε2µ [Λs, h3]∂x(T−1(h2bxxuux)), Λsux.
and thus, after remarking that
|∂x(T−1∂x(h3uxux))|Hs−1 ≤ |T−1∂x(h3uxux)|Hs
≤ k T−1∂
xkHs(R)→Hs(R)|h3uxux|Hs.
we can proceed as for the control of B3 to get
|B5| ≤ εC(Es(U ))Es(U )2.
• Estimate of (ΛsB(U ), SΛsU ). Note first that
B(U ) = εbxu T−1q2(U ) where, q2(·) as in (8), so that (ΛsB(U ), SΛsU ) = (Λs(εbxu), Λsζ) + (Λs(T−1q2(U )), TΛsu) = (Λs(εbxu), Λsζ) − [Λs, T]T−1q2(U ), Λsu + Λsq2(U ), Λsu.
Using again here the explicit expressions of T, q2(U ) and Lemma 2, we get
• Estimate of (Λsu, [∂
t, T]Λsu). We have that
(Λsu, [∂t, T]Λsu) = (Λsu, ∂thΛsu) + µ 3(Λ su x, ∂th3Λsux) −εµ2 (Λsu, ∂th2bxΛsux) − εµ 2 (Λ su x, ∂th2bxΛsu) +ε2µ(Λsu, ∂thb2xΛsu).
Controlling these terms by εC(Es(U ), |∂
th|L∞)Es(U )2follows directly from a
Cauchy-Schwarz inequality and an integration by parts.
Gathering the informations provided by the above estimates and using the fact that Hs(R) ⊂ W1,∞, we get
eελt∂t(e−ελtEs(U )2) ≤ ε C(Es(U ), |∂th|L∞) − λEs(U )2+ εC(Es(U ))Es(U ).
Taking λ = λT large enough (how large depending on supt∈[0,T ε]C(E
s(U ),|∂ th|L∞)
to have the first term of the right hand side negative for all t ∈ [0,Tε], one deduces
∀t ∈ [0,T
ε], e
ελt∂
t(e−ελtEs(U )2) ≤ εC(Es(U ))Es(U ).
Integrating this differential inequality yields therefore ∀t ∈ [0,T ε], E s(U ) ≤ eελTtEs(U 0) + ε Z t 0 eελT(t−t′)C(Es(U )(t′))dt′. 2.3. Main result. In this subsection we prove the main result of this paper, which shows well-posedness of the Green-Naghdi equations over large times.
Theorem 1. Let b ∈ Cb∞(R), t0 > 1/2, s ≥ t0+ 1. Let also the initial condition
U0 = (ζ0, u0)T ∈ Xs, and satisfy (3). Then there exists a maximal Tmax > 0,
uniformly bounded from below with respect to ε, µ ∈ (0, 1), such that the Green-Naghdi equations (2) admit a unique solution U = (ζ, u)T ∈ Xs
Tmax with the initial
condition (ζ0, u0)T and preserving the nonvanishing depth condition (3) for any
t ∈ [0,Tmax
ε ). In particular if Tmax< ∞ one has
|U(t, ·)|Xs−→ ∞ as t −→ Tmax,
or
inf
R h(t, ·) = infR 1 + ε(ζ(t, ·) − b(·)) −→ 0 as t −→ Tmax.
Morever, the following conservation of energy property holds
∂t |ζ|22+ (hu, u) + µ(hT u, u) = 0, where T = T [h, εb].
Remark 3. For 2D surface waves, non flat bottoms, B. A. Samaniego and D. Lannes [3] proved a well-posedness result to the Green-Naghdi using a Nash-Moser scheme. Our result only use a standard Picard iterative and there is therefore no loss of regularity of the solution with respect to the initial condition. In the one-dimensional case and for flat bottoms, our result coincides with the one proved by Li in [14].
Remark 4. Our approach does not admit a straightforward generalization to the 2D case. The main reason is that the natural energy norm Xs is then given by
|U|2
Xs = |ζ|2Hs+ |u|2Hs+ µ|∇ · u|2Hs,
which does not control the H1(R2) norm of u (since u takes its values in R2, the
information on the rotational of u is missing).
Remark 5. No smallness assumption on ε nor µ is required in the theorem. The fact that Tmax is uniformly bounded from below with respect to these parameters
allows us to say that if some smallness assumption is made on ε, then the existence time becomes larger, namely of order O(1/ε). This is consistent with the existence time obtained for the (simpler) physical models derived under some smallness as-sumption on ε, like the Boussinesq models. In fact, such models can be derived from the Green-Naghdi equations [13]. The present theorem also has some direct implication for the justification of variable-bottom Camassa-Holm equations [9]. Proof. We want to construct a sequence of approximate solution (Un= (ζn, un))
n≥0
by the induction relation
(17) U0= U0, and ∀n ∈ N,
∂
tUn+1+ A[Un]∂xUn+1+ B(Un) = 0;
U|n+1t=0 = U0.
By Proposition 1, we know that there is a unique solution Un+1 ∈ C([0, ∞); Xs)
to (17) if Un∈ C([0, ∞); Xs) and Un satisfies (3) for all times. Let R > 0 be such
that Es(U0) ≤ R/2, it follows from Proposition 1 that Un+1satisfies the following
inequality
Es(Un+1(t)) ≤ eελTtEs(U0) + ε
Z t
0
eελT(t−t′)C(Es(Un(t′))dt′,
we suppose now that
sup t∈[0,T ε] Es(Un(t)) ≤ R, therefore Es(Un+1(t)) ≤ R/2 + (eελTt− 1)(R/2 +C(R) λT ). Hence, there is T > 0 small enough such that
sup
t∈[0,T ε]
Es(Un+1(t)) ≤ R.
Using now the link between Es(U ) and |U|
Xs given by Lemma 3 we get
sup t∈[0,T ε] |Un+1(t)|Xs ≤ C 1 h0 R. We also know from the equations that
∂tζn+1= −hnun+1x − εζxnun+1+ εbxun+1.
Hence, one gets
(18) |∂thn+1|L∞ = ε|∂tζn+1|L∞ ≤ εC 1 h0 R. Since moreover hn+1= hn+1t=0 + Z t 0 ∂tζn+1,
we can deduce from (18) and the fact that hn+1t=0 = 1 + ε(ζ0− b) ≥ h0 that it is
possible to choose T small enough for Un+1to satisfy (3) on [0,T
ε], with h0replaced
by h0/2.
Finally, we deduce that the Cauchy problem
∂tUn+1+ A[Un]∂xUn+1+ B(Un) = 0;
U|n+1
t=0 = U0
has a unique solution Un+1satisfing (3) and the inequality
Es(Un+1) ≤ eελTtEs(U 0) + ε Z t 0 eελT(t−t′)C(Es(Un)(t′))dt′, when 0 ≤ t ≤ T
ε and λT depending only on supt∈[0,T ε]E
s(Un). Thanks to this
energy estimate, one can conclude classically (see e.g. [1]) to the existence of Tmax= T (Es(U0)) > 0,
and of a unique solution U ∈ Xs
Tmax to (2) preserving the inequality (3) for any
t ∈ [0,Tmax
ε ] as a limit of the iterative scheme
U0= U0, and ∀n ∈ N,
∂
tUn+1+ A[Un]∂xUn+1+ B(Un) = 0;
U|n+1
t=0 = U0.
The fact that Tmax is bounded from below by some T > 0 independent of ε, µ ∈
(0, 1) follows from the analysis above, while the behavior of the solution as t → Tmax
if Tmax< ∞ follows from standard continuation arguments.
Though the conservation of the energy can be found in some references (e.g. [5]), we reproduce it in Appendix B for the sake of completeness.
Appendix A. Existence of solutions for the linearized equations In this section we examine existence, uniqueness, and regularity for solutions to the following system of equations:
(19)
∂tU + A[U ]∂xU = f ;
U|t=0 = U0,
where U = (ζ, u)T ∈ Xs
T is such that ∂tU ∈ XTs−1 and satisfy the condition (3) on
[0,T
ε]. We begin the proof by the following lemma (see for instance [17]):
Lemma 4. Let ϕ ∈ C0∞(R), such that ϕ(r) = 1 for |r| ≤ 1. Let also Jδ = ϕ(δ|D|), δ > 0.
Then:
(i) ∀s, s′∈ R, Jδ: Hs(R) 7−→ Hs′
(R) is a bounded linear operator.
(ii) Jδ commutes with Λs and is self-adjoint operator.
(iii) ∀f ∈ C1(R) ∩ L∞(R), v ∈ L2(R) there exists C independent of δ such that
|[f, Jδ]v|H1≤ C|f|C1|v|L2.
(iv) ∀f ∈ Hs(R), s > 1
2, Jδf ∈ L∞(R) with
|Jδf |L∞≤ C|f|L∞
Our strategy will be to obtain a solution to (19) as a limit of solutions Uδ to
(20)
∂tUδ+ JδA[U ]Jδ∂xUδ = f ;
Uδ|t=0 = U0.
For any δ > 0, Aδ = JδA[U ]Jδ is a bounded linear operator on each Xs, and Fδ=
Aδ∂
x+B(U )∈ C1(Xs) so by Cauchy-Lipschitz the ODE (20) has a unique solution,
Uδ ∈ C([0, T/ε], Xs). Our task will be to obtain estimates on Uδ, independent of
δ ∈ (0, 1) and to show that the solution Uδ has a limit as δ ց 0 solving (19). To
do this, we remark that 1 2∂tE s(U δ)2 = −(SA[U]Λs∂xJδUδ, ΛsJδUδ) − Λs, A[U ]∂xJδUδ, SΛsJδUδ +(Λsf, SΛsUδ) + 1 2(Λ su δ, [∂t, T]Λsuδ) + S, JδΛsU δ, A[U ]Λs∂xJδUδ + S, JδΛsUδ,Λs, A[U ]∂xJδUδ. (21)
Note that we do not give any details for the control of the components of the r.h.s (21) other than the last two terms because the others can be handled exactly as in Proposition 1. To estimate the last two terms of the r.h.s (21), we have that
S, JδΛsU δ, A[U ]Λs∂xJδUδ = T, JδΛsu δ, T−1(hΛs∂xJδuδ) + T, JδΛsu δ, εuΛs∂xJδuδ + T, JδΛsu δ, T−1Q1[U ]Λs∂xJδuδ.
One can check by using the explicit expression of T that T, Jδ = h, Jδ −µ 3∂xh 3, Jδ∂ x− εµ 2 h 2b x, Jδ∂x +εµ 2 ∂xh 2b x, Jδ + ε2µhb2x, Jδ.
One deduces directly from Lemma 4, an integration by parts, the Cauchy-Schwarz inequality and the explicit expression of Q1[U] that
S, JδΛsU
δ, A[U ]Λs∂xJδUδ ≤ CEs(Uδ)2,
similarly, one can conclude S, JδΛsU
δ,Λs, A[U ]∂xJδUδ ≤ CEs(Uδ)2,
where C is a constant independent of δ. By the Proposition 1, we have −(SA[U]Λs∂xJδUδ, ΛsJδUδ) − Λs, A[U ]∂xJδUδ, SΛsJδUδ +(Λsf, SΛsUδ) +1 2(Λ su δ, [∂t, T]Λsuδ) ≤ CEs(Uδ)2+ CEs(f )2.
Consequently, we obtain an estimate of the form
(22) d
dtE
s(U
Thus Gronwall’s inequality yields an estimate (23) Es(Uδ)2≤ C(t)Es(U0)2+ sup
[0,t]
Es(f )2,
independent of δ ∈ (0, 1). Thanks to this energy estimate, one can conclude clas-sically (see e.g. [17]) to the existence of a unique solution U ∈ C([0, T ], Xs) to
(19).
Appendix B. Conservation of the energy In order to prove that
∂t |ζ|2 2+ (hu, u) + µ(hT u, u) = 0,
we multiply the first equation of (2) by ζ and the second by u, integrateon R, and sum both equations to find
1 2∂t|ζ|
2
2+(∂x(hu), ζ)+(∂tu, hu)+µ(hT ∂tu, u)+(∂xζ, hu)+ε(u∂xu, hu)+µε(Qu, u) = 0.
Therefore 1 2∂t|ζ| 2 2+ 1 2∂t(hu, u) − 1 2(∂th, u 2) + ε(u∂ xu, hu) + µ(hT ∂tu, u) + µε(Qu, u) = 0,
where the term Qu is defined as: Qu = −13∂x[(h3(u∂x2u − (∂xu)2) +ε
2[∂x(h
2u∂
x(u∂xb) − h2∂xb(u∂x2u − (∂xu)2)]
+ε2h∂xb(u∂x(u∂xb)).
Using now the fact that h = 1 + ε(ζ − b) and the first equation of (2), we get −12(∂th, u2) + ε(u∂xu, hu) = − ε 2(∂x(hu), u 2) + ε(u∂ xu, hu) = 0. Thus, (24) 1 2∂t|ζ| 2 2+ 1 2∂t(hu, u) + µ(hT ∂tu, u) + µε(Qu, u) = 0. Regarding now the term µ(hT ∂tu, u), we remark as in [5] that
µ(hT ∂tu, u) = µ(T1∗hT1∂tu, u) + µ(T2∗hT2∂tu, u),
= µ(hT1∂tu, T1u) + µ(hT2∂tu, T2u),
= µ(h(∂t(T1u) − ∂tT1u), T1u) + µ(h∂t(T2u), T2u),
with Tj∗(j = 1, 2) denoting the adjoint of the operators Tj given by
T1u = h √ 3∂xu − ε √ 3 2 ∂xbu, and T2u = ε 2∂xbu. It comes: µ(hT ∂tu, u) = µ 2∂t(hT1u, T1u) − µ 2(∂th, (T1u) 2 ) − µ(h(∂tT1u, T1u) +µ 2∂t(hT2u, T2u) − µ 2(∂th, (T2u) 2), = µ 2∂t(hT1u, T1u) + µ 2∂t(hT2u, T2u) − µ 2(∂th, (T1u) 2 + (T2u)2) −µ(h∂tT1u, T1u).
Inject this result in (24) to get: 1 2∂t |ζ|2 2+ (hu, u) + µ(hT u, u) = µ 2(∂th, (T1u) 2+ (T 2u)2) +µ(h∂tT1u, T1u) − µε(Qu, u). Noting that h∂tT1u = ∂th(T1u + √ 3T2u), it comes: 1 2(∂th, (T1u) 2 + (T2u)2) + (h∂tT1u, T1u) = 1 2 ∂th, 3(T1u)2+ (T2u)2+ 2 √ 3T1uT2u , = µ 2 ∂th, ( √ 3T1u + T2u)2 , = εu,h 2∂x( √ 3T1u + T2u)2 , where we used here the first equation of (2). Finally, we get:
1 2∂t |ζ|22+ (hu, u) + µ(hT u, u) = µεh 2∂x( √ 3T1u + T2u)2− Qu, u . One can easily show thath
2∂x( √
3T1u + T2u)2− Qu, u
= 0, which implies easily the result.
Acknowledgments. The author is grateful to David Lannes for encouragement and many helpful discussions.
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Universit´e Bordeaux I; IMB, 351 Cours de la Lib´eration, 33405 Talence Cedex, France
